Since the commencement of their development over four decades ago for use as surgical sutures, the family of absorbable polyesters has commanded a significant role as useful or potentially useful biomaterials for use in many medical and pharmaceutical applications. However, the existing and consistently growing applications of absorbable polyesters have been limited to a small number of polymers comprising linear chains, which (1) are usually melt-processable; (2) have limited solubility in common organic solvents; (3) are difficult to produce with a high degree of purity or in monomer-free forms, which can be critical to their successful use in certain biomedical applications; and (4) are practically unsuitable for producing high-compliance, resilient microfibers or films while retaining a minimum degree of crystallinity required for achieving dimensional stability and surface hardness. To address most of these shortcomings of linear polyesters, while reaching a new milestone in the design of absorbable polymers with exceptionally high degrees of freedom for molecular chain tailoring to meet unique requirements of contemporary medical devices in new as well as unfulfilled applications, the present inventor has introduced and implemented the concept of crystalline segmented, polyaxial copolyester chains comprising a monocentric, amorphous, highly compliant polyaxial core, end-grafted with crystallizable segments (U.S. Pat. Nos: 6,462,169; 6,794,485; 7,070,858; and 7,129,319). Success associated with the exceptional properties of this family of polyaxial segmented copolyesters provided a strong incentive to address further the linear polyester shortcomings, including those associated with polymer purity and solubility and to pursue the study subject of the present invention pertaining to precisely crafted polymer properties toward reaching a second milestone in the medical application of absorbable polyesters.